论文标题
通过预测和正常截面
Geometry of surfaces in $\mathbb R^5$ through projections and normal sections
论文作者
论文摘要
我们通过将$ \ Mathbb r^5 $的表面几何形状与正交预测获得的$ \ Mathbb r^4 $中的常规和奇异表面的几何形状联系起来。特别是，我们获得了渐近方向之间的关系，这不是$ \ Mathbb r^5 $中表面的二阶几何形状，但属于$ \ Mathbb r^4 $。我们还将每种表面的脐带曲率及其与球形接触。然后，我们将表面视为$ \ mathbb r^6 $的3个manifolds的正常部分，并通过定义适当的脐带曲率的3个manifolds，再次将渐近方向和与球接触。
We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in $\mathbb R^5$ but are in $\mathbb R^4$. We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in $\mathbb R^6$ and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.
